# Expression of finitely generated ideal of algebra

Suppose $$\mathcal A$$ is $$R$$-algebra, and $$\alpha_1, \alpha_2, \cdots , \alpha_n \in \mathcal A$$.

Two-sided ideal of $$\mathcal A$$ generated by $$\alpha_1, \alpha_2, \cdots , \alpha_n$$ is

$$\left\{ \beta_1 \alpha_1 + \beta_2 \alpha_2 + \cdots + \beta_n \alpha_n: \beta_1, \beta_2, \cdots, \beta_n \in \mathcal A \right\}$$

I am solving this problem. But to show this set is right ideal of $$\mathcal A$$, I think condition of commutativity of $$\mathcal A$$ is necessary.

• What i your question? – PrudiiArca 2 days ago
• I am not used to noncommutative algebra, so this is not an answer. I think it would suffice if the $a_i$ commute with any element of $A$, so they are from some sort of center ideal (without taking notes it seems to me like that these elements form an ideal, I am not certain though) – PrudiiArca 2 days ago